The <Dambis-Dubins-Schwarz theorem> says that there is Brownian motion $W$ such that
$$
M_t=W_{[M]_t}.
$$
When $[M]$ is strictly increasing, define its inverse
$$
T_s=\inf\{t:[M]_t>s\}
$$
and set $W_s=M_{T_s}$. Optional sampling shows that $W$ is a continuous local martingale, while time change gives $[W]_s=s$. The <Lévy characterization of Brownian motion> makes $W$ Brownian, and inverse time change gives the displayed representation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-202/4/b/solution